On gravitational quantum tunnelling

Question

All celestial bodies have a gravitational well, and particle in the vicinity of this well would feel a gravitational force. My questions are:

a)How can I find the thickness of such a gravitational well?

b)Shouldn't the subatomic particles, e.g. An electron be confined to the potential well of a planet or a star?

c)Is quantum mechanical tunnelling a possibility for the electron (here) to get over the potential barrier?

E.g.: A black hole of infinite mass in the presence of another body becomes completely transparent quantum mechanically ( $\Pi$ = 1). But in an Aharonov-Bohm-like effect, if we consider two systems, each with black hole (B.H.) and concentric shell, opposite each other can result in a tunneling probability, $\Pi$ greater than 0.

In a simplified model of a black hole facing a body of mass $M_{2}$. $M_{2}$ is centered at $R$ opposite a black hole of mass $M$ centered at the origin. Since tunneling is greatest near the top of the barrier, the deviation from a $\frac{1}{r}$ potential toward the center of each body is not critical. The potentials used are that of two point masses, so $M_{2}$ may also be a black hole. Thus two little black holes may get quite close for maximum tunneling radiation. Solving the Schrödinger equation outside the black hole:

$$-\frac{\hbar^{2}}{2m}{D^{2}\psi}=-\left[\frac{-GmM}{r}+\left(-\frac{GmM_{2}}{R-r}\right)-E\right]\psi$$ In the region $a\leq r\leq b$, where $a$ bad $b$ are classical turning points, and $$E=-\frac{GmM}{a}+\left(\frac{-GmM}{R-a}\right)=-\frac{GmM}{b}+\left(\frac{-GmM_{2}}{R-b}\right)$$ Since, $T=e^{-2\Delta \gamma}$, $$\Delta \gamma=\frac{m}{\hbar}\sqrt{\frac{2GM}{d}}\left[\sqrt{b(b-d)}-\sqrt{a(a-d)}-dln\left|\frac{\sqrt{b}+\sqrt{b-d}}{\sqrt{a}+\sqrt{a-d}}\right|\right]$$ Here $d=\frac{Ma(R-a)R}{[M((R-a)R+M_{2}a^{2}]}$ and for $R>>b$ & $M_{2}>>M$

Thus, $\Delta \gamma$ approaches zero as $a$ approaches $b$, yielding $\Pi$ approaches 1. When $M$ approaches zero, or $M_{2}$ approaches infinity, or equivalently $\frac{M}{M_{2}}$ approaches zero, $\Delta \gamma$ approaches zero and $\Pi$ approaches one. Hence observing quantum tunnelling.

For a better detailed derivation refer here

Answer 1

Quantum tunneling enables the penetration of a particle through a potential barrier without the expenditure of energy. It requires that you have the same energy available on both sides of the barrier. In the gravitational "potential well" there is no potential barrier that can be penetrated. Even at infinity the potential is higher than any energy state in the interior of the well.

A gravitational potential barrier exists between two close celestial bodies, like between the moon and the earth. However, these barriers are so extremely thick that a tunneling probability for a subatomic particle can be excluded.

However, a subatomic particle, like an electron, can, due to its low mass, acquire such a high thermal velocity, the escape velocity $$v=\sqrt{\frac{2GM}{r}}$$ that it can (classically) leave the potential well of a celestial body. For the earth at sea level the escape velocity is $11.2km/s$. The mean thermal velocity of an electron at room temperature is $75km/s$. Thus an electron with mean thermal velocity, if uninhibited, would classically leave the potential well of the earth.

Answer 2

I was asking myself the same question. Upon further thinking I had concluded that yes it should be possible for an object to tunnel 'out' of such a gravity potential.

As others have pointed out, the gravitational potential well is not a super well defined well. The potential energy increases non-linearly to distance. But this should not mean that tunnelling it is not possible.

One could just simplify the problem into a 2-dimensional gravitational force vs distance problem. In this case the gravitational attractive force must be defined as a function of distance. A Hamiltonian can then be constructed to represent the potential energy of the objects as a function of distance. A particle-like localised wavefunction representing the probability of the particle being found can then be appropriately defined. Then, to see what happens to such a particle over time, one can plot the wavefunction over a range of time (present to future) by letting it evolve over time according to the Hamiltonian.

Answer 3

How can I find the thickness of such a gravitational well?

Theoretically you shouldn't, the effect of the mass should propagate for ever. Practically, it depends on how sensitive your measuring instruments are.

Shouldn't the subatomic particles, e.g. An electron be confined to the potential well of a planet or a star?

Why? You refer yourself in your title to quantum tunnelling, which is usually used in connection with electromagnetic forces, but it's not exclusively confined to that force. Although, the heavier the mass, the less likely you are to find a particle relatively far away from it. But I think you might be mixing up electrostatic and gravitational wells when you use the word tunnelling in the context of this question.

If there is a well defined thickness, then is tunnelling a possibility for the electron (here) to get over the potential barrier?

There is no abrupt cut off point in these situations. No well defined thickness is involved. Again, I would not see it in conventional tunnelling terms.